Beamformer phase optimization for a multi-layer mimo system augmented by radio distribution network

ABSTRACT

A system for selecting optimal phase combinations for RF beamformers in a MIMO hybrid receiving systems augmented by RF Distribution Network. The system addresses the issue of providing beamforming gains for a plurality of layers using one common set of weights for each beamformer. The specification may be based on channel estimation of all layers as viewed by all receiving antennas, and maximizing metrics that capture the total received power.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation application of U.S. non-provisionalpatent application Ser. No. 13/776,204 filed on Feb. 25, 2013, which isa continuation-in-part application of U.S. non-provisional patentapplication Ser. No. 13/630,146 filed on Sep. 28, 2012, which in turnclaims benefit from U.S. provisional patent application Nos. 61/652,743filed on May 29, 2012; 61/657,999 filed on Jun. 11, 2012; and 61/665,592filed on Jun. 28, 2012; and U.S. non-provisional patent application Ser.No. 13/776,204 further claims benefit from U.S. provisional patentapplication Nos. 61/658,015 filed on Jun. 11, 2012; 61/658,010 filed onJun. 11, 2012; 61/658,012 filed on Jun. 11, 2012; and 61/671,416 filedon Jul. 13, 2012, all of which are incorporated herein by reference intheir entirety.

FIELD OF THE INVENTION

The present invention relates generally to the field of radio frequency(RF) multiple-input-multiple-output (MIMO) systems and more particularlyto systems and methods for RF MIMO systems using RF beamforming and/ordigital signal processing, to augment the receiver performance.

BACKGROUND

Prior to setting forth a short discussion of the related art, it may behelpful to set forth definitions of certain terms that will be usedhereinafter.

The term “MIMO” as used herein, is defined as the use of multipleantennas at both the transmitter and receiver to improve communicationperformance. MIMO offers significant increases in data throughput andlink range without additional bandwidth or increased transmit power. Itachieves this goal by spreading the transmit power over the antennas toachieve spatial multiplexing that improves the spectral efficiency (morebits per second per Hz of bandwidth) or to achieve a diversity gain thatimproves the link reliability (reduced fading), or increased antennadirectivity.

The term “beamforming” sometimes referred to as “spatial filtering” asused herein, is a signal processing technique used in antenna arrays fordirectional signal transmission or reception. This is achieved bycombining elements in the array in such a way that signals at particularangles experience constructive interference while others experiencedestructive interference. Beamforming can be used at both thetransmitting and receiving ends in order to achieve spatial selectivity.

The term “beamformer” as used herein refers to RF circuitry thatimplements beamforming and usually includes a combiner and may furtherinclude switches, controllable phase shifters, and in some casesamplifiers and/or attenuators.

The term “Receiving Radio Distribution Network” or “Rx RDN” or simply“RDN” as used herein is defined as a group of beamformers as set forthabove.

The term “hybrid MIMO RDN” as used herein is defined as a MIMO systemthat employs two or more antennas per channel (N is the number ofchannels and M is the total number of antennas and M>N). Thisarchitecture employs a beamformer for each channel so that two or moreantennas are combined for each radio circuit that is connected to eachone of the channels.

In hybrid MIMO RDN receiving systems, when the phases of the receivedsignals from each antenna are properly adjusted or tuned with respect toone another, the individual signals may be combined and result in animproved performance of the receiving system.

FIG. 1 shows an example of a standard 2×2 MIMO radio 20 with twoantennas A and B communicating with a base station 10 having twotransmit antennas radiating Tx1 and Tx2. While each antenna receivesboth transmitted layers, the baseband separates them and processes themin an optimal way

SUMMARY

Embodiments of the present invention address the challenge of aligningthe phases in the receive antennas coupled to the beamformers in thehybrid MIMO RDN architecture, in order to mitigate the combiners lossescaused by misaligned phases.

Embodiments of the present invention are based on seeking maximizationof the total power received from all transmitted layers as measured bythe MIMO's baseband; the summation includes all transmitting antennassignals, as viewed by all receiving RDN antennas, which are equippedwith phase shifters.

The received powers may be measured via channel estimation of individualantennas thru their respective beamformers, radios and basebandcircuitry.

Different metrics are provided to quantify the said total receivedpower:

$\mspace{20mu} {P_{Total} = {\text{?}\text{?}{\sum\limits_{k = 1}^{L}{\text{?}\text{?}P_{j,k}S_{j,k}}}}}$  P_(j, k) = [abs(S_(j, k))]²   j = 1, 2  …  M,   k = 1, 2  …  L?indicates text missing or illegible when filed

It would be therefore advantageous to find a way to use a single degreeof freedom i.e. the need to choose or select one phase in aligning abeamformer that serves 2, 4, or more different phase setting, stemmingfrom the fact that multiple incoming signals have each a specificpossible phase alignment for the beamformer.

The requirement for optimal alignment of phases for all transmittedlayers appears also in higher MIMO ranks and in various RDNconfigurations. A general optimization process is addressed inembodiments of the invention described herein.

BRIEF DESCRIPTION OF THE DRAWINGS

For a better understanding of the invention and in order to show how itmay be implemented, references are made, purely by way of example, tothe accompanying drawings in which like numerals designate correspondingelements or sections. In the accompanying drawings:

FIG. 1 is a high level block diagram illustrating a system according tosome embodiments of the prior art;

FIG. 2 is a high level block diagram illustrating a system according tosome embodiments of the present invention;

FIGS. 3 and 4 are signal diagrams illustrating an aspect according toembodiments of the present invention;

FIG. 5 is a table with signal diagrams illustrating an aspect accordingto embodiments of the present invention; and

FIG. 6 is a signal diagram illustrating yet another aspect according toembodiments of the present invention.

The drawings together with the following detailed description make theembodiments of the invention apparent to those skilled in the art.

DETAILED DESCRIPTION

With specific reference now to the drawings in detail, it is stressedthat the particulars shown are for the purpose of example and solely fordiscussing the preferred embodiments of the present invention, and arepresented in the cause of providing what is believed to be the mostuseful and readily understood description of the principles andconceptual aspects of the invention. In this regard, no attempt is madeto show structural details of the invention in more detail than isnecessary for a fundamental understanding of the invention. Thedescription taken with the drawings makes apparent to those skilled inthe art how the several forms of the invention may be embodied inpractice.

Before explaining the embodiments of the invention in detail, it is tobe understood that the invention is not limited in its application tothe details of construction and the arrangement of the components setforth in the following descriptions or illustrated in the drawings. Theinvention is applicable to other embodiments and may be practiced orcarried out in various ways. Also, it is to be understood that thephraseology and terminology employed herein is for the purpose ofdescription and should not be regarded as limiting.

FIG. 2 shows an example of a 2×2 MIMO RDN architecture in which eachreceive antenna as shown in FIG. 1 such as A1, and B1 are enhanced byadding another antenna, A2 and B2 respectively, thus providing receptionby four antennas instead of two. The hybrid MIMO RDN architecturefurther includes phase shifters 40-1 and 40-2 and combiners 30-1 and30-2.

Without losing generality, and for the sake of simplified explanation,it is assumed herein that the base station transmits each layer over oneTx antenna.

The Hybrid MIMO RDN can provide an additional gain however, as thecombiners 30-1 and 30-2 are serving two different Tx antennas with onlyone phase shifter, it is possible that the diversity parameters (e.g.,phase) that are used to optimize the reception of Tx1 are not the sameas those needed for receiving Tx2. This is especially true if theantennas are not correlated from one to another.

As seen in FIG. 2, if the phase shift introduced in the path fromantenna radiating Tx1 (1LA1-1LA2) is compensated by the phase shifter,that phase shifter setting will only be correct it the paths from theantenna radiating Tx2 are the same. That is, the phase setting will onlybe correct if (2LA1-2LA2) is the same or a multiple wavelength from(1LA1-1LA2). A similar outcome holds for the Tx1 and Tx2 signalsreceived by Antennas B1 and B2.

If the case using four 90° phase shifts is compared to align the signalsfrom Tx1, it is apparent that there are three possible outcomes for theTx2 signal:

The first outcome is that the signals arrive at the antennas A1 and A2with a similar phase differences as for the Tx1 transmission so the samephase setting used to enhance the reception of Tx1 will also enhanceTx2. (25%);

The second outcome is that the resulting Tx2 signals to A1 and A2 are+/−90° from each other and will produce zero diversity gain for thisprocess. (50%); and

The third outcome is that the resulting Tx2 signals are 180° from eachother and can cancel each other or produce a negative diversity gaindepending on their relative amplitudes. (25%).

When the result is the aforementioned third outcome, the system mustchoose to sacrifice diversity gain for Tx1 in order to avoid the totalloss of the Tx2 signal. This would result in low diversity gain (−0 dB)for both Tx1 and Tx2.

The algorithm offered by embodiments of the invention results in phaseoptimization based on seeking maximization of the total power receivedfrom all transmitted layers as measured by the MIMO's baseband; thesummation includes all transmitting antennas signals, as viewed by allreceiving RDN antennas, which are equipped with phase shifters. Theaforementioned received powers are measured via channel estimation ofindividual antennas thru their respective beamformers, radios andbaseband circuitry.

In accordance with some embodiments of the present invention, a multipleinputs multiple outputs (MIMO) receiving system having number N channelsis provided. The MIMO receiving system may include a radio distributednetwork (RDN) having number N beamformers, each having number K_(N)antennas. The MIMO system may further include at least one phase shifterassociated with one or more of the N beamformers. Additionally, the MIMOreceiving system is configured to: (a) select one phase that optimizesperformance of multiple layers, via channel estimation of each layer asseen (e.g, taking into account the gain and phase affected by thephysical location) by each receiving antenna, and (b) maximize a totalreceived power from all transmitted signals.

FIGS. 3 and 4 are signal diagrams illustrating an aspect according toembodiments of the present invention. In the following non-limitingexample, a case of N plurality of uncorrelated transmit signalsprojected from a base station, where N=2, is received by a 2×2 MIMO UEwhich is augmented by an RDN with 2 beamformers, each beamformer hasthree receive antennas. It is assumed, for the sake of the followingexample that the beamformers can select one of 4 possible phases: 0°,90°, 180°, and 270°. When selecting the beamformers' phases is such away that will maximize the received signal coming from Tx1, the Tx 2phases may or may not be constructively combined, but rather, may have1*4*4=16 phase combinations.

For the sake of simplicity, it is assumed that each receive antennaprovides the same amplitude and a randomly selected phase out of 4alternatives. It is also assumed that the amplitudes power is 0.33 (forthe sake of the example).

As the signals are fed into an RF combiner, the translation into voltageof each signal provides a combined result as described herein below:

In FIG. 3-A, 3 aligned vectors, each of ⅓ of a Watt are depicted so thatthe combined voltage equals 3×SQRT (0.333)=1.732 V=>3 W. Since the gainis the output divided by the input, the gain here equals 3 W/1 W=3,hence 4.77 dB. FIG. 3-B depicts two aligned vectors each of ⅓ of a Wattand a single ⅓ Watt perpendicular vector so that the combined voltageequals SQRT (0.33)+j SQRT (0.33)=>Effective voltage combining=SQRT[(2×0.577)²+0.577²]=>1.67 W. For a similar reason, the gain equals 1.67W/1 W=1.67 hence 2.2 dB. Similar calculation for FIGS. 3-C, D, E, F, Ggenerates the gains of 2.2 for each one. Applying similar calculationfor FIGS. 3-H, I, J, K, L, M, N, O, P generates the gains of −4.77 dBfor each one.

As can be seen, the seven combinations described in FIG. 3-A through Gprovide positive gains for both layers, while the nine combinationsdescribed by FIG. 4-H through P, provide positive gain for one layer andnegative gain for the other.

FIG. 5 demonstrates in form of a table and a corresponding signaldiagram how theses conclusion have been reached. Specifically, table 500illustrates the aforementioned calculations with specific configurations501-505.

It can be easily seen that while configuration 501 yields 4.77 dB gain,configuration 502 yields a lesser yet still positive gain of 2.22 dB.Configurations 503-505 on the other hand, yield a negative gain of −4.77dB.

As can seen above in FIGS. 3 and 4 when aligning one transmit signal tobest gain, the second transmit signal is left for random combination ofphases, and may become exceedingly adverse at many of the cases.

FIG. 6 illustrates improvements that can be achieved by embodiments ofthe present invention in overall gain terms. The upper part illustratescases where the selection of a maximal gain for Tx 1, where all antennasare aligned, undermines the gain for Tx 2. The lower part of FIG. 6demonstrates that replacing +4.77 dB gain for layer 1 and −4.77 dB forlayer 2 provides gains of +2.22 dB for both layers at 9 out of 16 of thecases; (it is noted that in other 6 cases, the corresponding gains are+4.77 dB and 2.22 dB, and in one other case both are +4.77 dB)

Similar approaches can be applied to more complex MIMO hybrid RDNconfigurations, where there are more layers and or more antennas arecombined by RF beamformers.

One embodiment of metrics and a procedure for the selection of optimalphase settings to all participating beamformers is described below:

Consider a beamformer with K_(N) receive antennas, each of themreceiving signals from M transmit antennas. The channel functionsh_(i,j,k) from transmit antenna j, j=1,2 . . . NM, to receive antenna i,i=1,2 . . . K_(N)N, at frequency k, k=1,2 . . . L (it is assumed thegeneral case of frequency selective channels) are obtained throughchannel estimation done by the base-band.

Each receive antenna is equipped with a set A of R phase shifters,}, forphase adjustment. The set A of phase shifters could be, for example, {0,90, 180, 270} degrees. The algorithm needs to select the optimal phaseφ_(i)∈A to be applied to receive antenna:

i, i=1,2 . . . K_(N), N

After phase adjusting the K_(N),N receive antennas, the overall channelfunctions seen by the receiver under consideration are:

  ? = ???, ?indicates text missing or illegible when filed

j=1,2 . . . NM, k=1,2 . . . L

A power is associated with each one of them:

? = [abs(?)]², ? = 1, 2??NM?k = 1, 2?…?L?indicates text missing or illegible when filed

In one embodiment the algorithm selects phases φ_(i)∈A, i=1,2 . . .K_(N), N, so as to maximize the total power P_(Total) defined as:

$\mspace{20mu} {P_{Total} = {\text{?}\text{?}{\sum\limits_{k = 1}^{L}{\text{?}{\text{?}.\text{?}}\text{indicates text missing or illegible when filed}}}}}$

In another embodiment, a procedure and metrics is provided wherein theantennas, e.g., antenna phases, are adjusted one by one recursively. Asbefore, φ₁ may be set to zero. To calculate φ₂ the contributions fromonly h_(1,j,k) and h_(2,j,k) are considered. The combined channels_(2,j,k) and channel power p_(2,j,k) for the first two antennas aredefined as:

s _(2,j,k) =h _(1,j,k) e ^(jφ) ¹ +h _(2,j,k) e ^(jφ) ² , j=1,2, . . . N,k=1,2 . . . L

p _(2,j,k) =[abs(s _(2,j,k))]² , j=1,2, . . . N, k=1,2 . . . L

The algorithm selects or chooses φ₂∈S that maximizes

$\sum\limits_{j = 1}^{N}{{\sum\limits_{k = 1}^{L}{\; {p_{2,j,k}.}}}}$

Continuing in a similar fashion for all antennas, once φ_(i−1) has beencalculated or determined, φ_(i) is calculated. Define:

${s_{i,j,k} = {\sum\limits_{l = 1}^{i}{h_{l,j,k}^{{j\Phi}_{l}}}}},{j = 1},{2\mspace{14mu} \ldots \mspace{14mu} N},{k = 1},{2\mspace{14mu} \ldots \mspace{14mu} L}$p_(i, j, k) = [abs(s_(i, j, k))]², j = 1, 2  …  N, k = 1, 2  …  L

Then, similarly, the algorithm selects or chooses φ_(i)∈S that maximizes

$\sum\limits_{j = 1}^{N}{{\sum\limits_{k = 1}^{L}{\; {p_{i,j,k}.}}}}$

The total number of possible antenna phase combinations for therecursive algorithm is R(K_(N)−1).

Since the order in which the antennas are optimized may affect theoutcome, some criterion may be used for numbering of the antennas. Forexample, in some embodiments the antennas may be sorted or ordered inascending/descending order based on the total power P_(Ant) _(i) of eachantenna:

${P_{{Ant}_{i}} = {\sum\limits_{j = 1}^{N}{\sum\limits_{k = 1}^{L}\left\lbrack {{abs}\left( h_{i,j,k} \right)} \right\rbrack^{2}}}},{i = 1},{2\mspace{14mu} \ldots \mspace{14mu} K_{N}}$

By repeating the aforementioned process for all beamformers in thehybrid MIMO RDN system an optimized overall gain for the entire hybridMIMO RDN architecture is achieved.

As will be appreciated by one skilled in the art, aspects of the presentinvention may be embodied as a system, method or an apparatus.Accordingly, aspects of the present invention may take the form of anentirely hardware embodiment, an entirely software embodiment (includingfirmware, resident software, micro-code, etc.) or an embodimentcombining software and hardware aspects that may all generally bereferred to herein as a “circuit,” “module” or “system.”

In various embodiments, computational modules may be implemented bye.g., processors (e.g., a general purpose computer processor or centralprocessing unit executing code or software), or digital signalprocessors (DSPs), or other circuitry. The baseband modem may beimplanted, for example, as a DSP. A beamforming matrix can be calculatedand implemented for example by software running on general purposeprocessor. Beamformers, gain controllers, switches, combiners, and phaseshifters may be implemented, for example using RF circuitries.

The aforementioned flowchart and block diagrams illustrate thearchitecture, functionality, and operation of possible implementationsof systems and methods according to various embodiments of the presentinvention. In this regard, each block in the flowchart or block diagramsmay represent a module, segment, or portion of code, which comprises oneor more executable instructions for implementing the specified logicalfunction(s). It should also be noted that, in some alternativeimplementations, the functions noted in the block may occur out of theorder noted in the figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts, or combinations of special purpose hardware andcomputer instructions.

In the above description, an embodiment is an example or implementationof the inventions. The various appearances of “one embodiment,” “anembodiment” or “some embodiments” do not necessarily all refer to thesame embodiments.

Although various features of the invention may be described in thecontext of a single embodiment, the features may also be providedseparately or in any suitable combination. Conversely, although theinvention may be described herein in the context of separate embodimentsfor clarity, the invention may also be implemented in a singleembodiment.

Reference in the specification to “some embodiments”, “an embodiment”,“one embodiment” or “other embodiments” means that a particular feature,structure, or characteristic described in connection with theembodiments is included in at least some embodiments, but notnecessarily all embodiments, of the inventions.

It is to be understood that the phraseology and terminology employedherein is not to be construed as limiting and are for descriptivepurpose only.

The principles and uses of the teachings of the present invention may bebetter understood with reference to the accompanying description,figures and examples.

It is to be understood that the details set forth herein do not construea limitation to an application of the invention.

Furthermore, it is to be understood that the invention can be carriedout or practiced in various ways and that the invention can beimplemented in embodiments other than the ones outlined in thedescription above.

It is to be understood that the terms “including”, “comprising”,“consisting” and grammatical variants thereof do not preclude theaddition of one or more components, features, steps, or integers orgroups thereof and that the terms are to be construed as specifyingcomponents, features, steps or integers.

If the specification or claims refer to “an additional” element, thatdoes not preclude there being more than one of the additional element.

It is to be understood that where the claims or specification refer to“a” or “an” element, such reference is not be construed that there isonly one of that element.

It is to be understood that where the specification states that acomponent, feature, structure, or characteristic “may”, “might”, “can”or “could” be included, that particular component, feature, structure,or characteristic is not required to be included.

Where applicable, although state diagrams, flow diagrams or both may beused to describe embodiments, the invention is not limited to thosediagrams or to the corresponding descriptions. For example, flow neednot move through each illustrated box or state, or in exactly the sameorder as illustrated and described.

The descriptions, examples, methods and materials presented in theclaims and the specification are not to be construed as limiting butrather as illustrative only.

Meanings of technical and scientific terms used herein are to becommonly understood as by one of ordinary skill in the art to which theinvention belongs, unless otherwise defined.

The present invention may be implemented in the testing or practice withmethods and materials equivalent or similar to those described herein.

While the invention has been described with respect to a limited numberof embodiments, these should not be construed as limitations on thescope of the invention, but rather as exemplifications of some of thepreferred embodiments. Other possible variations, modifications, andapplications are also within the scope of the invention. Accordingly,the scope of the invention should not be limited by what has thus farbeen described, but by the appended claims and their legal equivalents.

We claim:
 1. A multiple inputs multiple outputs (MIMO) receiving systemhaving number N channels, comprising: a radio distributed network (RDN)having number N beamformers, each having number K_(N) antennas; at leastone phase shifter associated with one or more of the N beamformers,wherein the MIMO receiving system is configured to: (a) select one phasethat optimizes performance of multiple layers, via channel estimation ofeach layer as seen by each receiving antenna, and (b) maximize a totalreceived power from all transmitted signals.
 2. The system according toclaim 1, wherein the MIMO receiving system optimizes a beamformer phaseby: performing channel estimation of each layer from each receivingantenna, selecting metrics that capture the combined received power, andusing relative phase setting between various antennas to calculate a setthat maximizes that metrics.
 3. The system according to claim 1, whereinphase optimization is carried out by: calculating channel functionsh_(i,j,k) from each one of the N transmit antenna j, j=1,2 . . . NM, toeach one of the K_(N) receive antenna i, i=1,2 . . . NK_(N), atfrequency k, k=1,2 . . . L, at the baseband module using channelestimation; selecting phases, wherein  A = ?(???, ϕ₁?…????indicates text missing or illegible when filedso as to maximize a total power P_(total) defined as:$\mspace{20mu} {{P_{Total} = {\text{?}\text{?}{\sum\limits_{k = 1}^{L}{\text{?}\text{?}}}}},{\text{?}\text{indicates text missing or illegible when filed}}}$wherein P_(j,k) denotes power associated with each one of receivedsignals s_(j,k) wherein  s_(j, k)?h_(i, j, k)^(jΦ_(i)), ?indicates text missing or illegible when filedi=1,2 . . . NM, k=1,2 . . . L so that? = [abs(?)]², ? = 1, 2??NM?k = 1, 2?…?L;?indicates text missing or illegible when filed and repeating thecalculating and the selecting stages for each one of the N beamformers.4. The system according to claim 1, wherein phase optimization iscarried out by: adjusting the antennas one by one recursively, whereinφ₁ is set to zero, and only contributions from and h_(1,j,k) andh_(2,j,k) are used to calculate φ₂; defining a combined channels_(2,j,k) and a channel power p_(2,j,k) for the first two antennas as:p_(2,j,k)=[abs(s_(2,j,k))]², j=1,2 . . . N, k=1,2 . . . L; and choosingφ₂∈S that maximizes$\sum\limits_{j = 1}^{N}{{\sum\limits_{k = 1}^{L}{\; {p_{2,j,k}.}}}}$5. The system according to claim 4, wherein the optimization of thephase of the beamformer is calculated wherein once φ_(i−1) has beendetermined, φ_(i) is calculated.
 6. The system according to claim 5,wherein:${s_{i,j,k} = {\sum\limits_{l = 1}^{i}{h_{l,j,k}^{{j\Phi}_{l}}}}},{j = 1},{2\mspace{14mu} \ldots \mspace{14mu} N},{k = 1},{2\mspace{14mu} \ldots \mspace{14mu} L}$p_(i, j, k) = [abs(s_(i, j, k))]², j = 1, 2  …  N, k = 1, 2  …  Land where in the algorithm chooses φ_(i)∈S that maximizes$\sum\limits_{j = 1}^{N}{{\sum\limits_{k = 1}^{L}{\; {p_{i,j,k}.}}}}$7. The system according to claim 5, wherein:  ?,   j = 1, 2  …  M, i,   i = 1, 2  …  N, k,   k = 1, 2  …  L  Φ_(i) ∈ A, i = 1, 2  …  N  A = [(ϕ]?ϕ₂  …  ϕ_(R)}P_(Total)$\mspace{20mu} {P_{Total} = {\text{?}\text{?}{\sum\limits_{k = 1}^{L}{\text{?}\text{?}P_{j,k}S_{j,k}}}}}$$\mspace{20mu} {S_{j,k} = {\sum\limits_{i = 1}^{N}{h_{i,j,k}^{{j\Phi}_{i}}}}}$  i = 1, 2  …  M,   k = 1, 2  …  L  P_(j, k) = [abs(S_(j, k))]²  j = 1, 2  …  M,   k = 1, 2  …  L.?indicates text missing or illegible when filed